Winter 2009 MATH 330B/539 Geometry Homework 6 Hints

Assigned Homework Set #6

Due Friday, February 20, 2009 at the start of class.
6.1 # 6, 9, 12, 15, 18

Hints for Assigned Problems

Remark on #6. The proof that uses Corollary 6.1 should be easy. The second proof that the author describes is a proof by contradiction. In the past, many of you have had trouble with the structure of proofs by contradiction. Be sure to work hard on this one.
Hint for #9: Consider three non-collinear points whose reflections are particularly easy to determine.
Hint for #12:
- How do you know that the result will be a glide reflection? (Hint: Find a theorem.)
- Pick one convenient point and consider its image.
  - Let point A be the intersection of lines L and M.
  - Let point B be the point on line K such that AB is perpendicular to line K.
  - Determine the image B'.
- Pick another convenient point and consider its image.
  - Draw a line N such that lines N, K, and L are equally spaced in that order.
  - Let point C be the intersection of lines K and M.
  - Let point D be the point on line N such that CD is perpendicular to line N.
  - Determine the image D'.
- What do you notice about points B, B', D, D'? What information does this give you about the Glide Reflection?
Hints for #15:
- Hint for (a): Given a composition of n reflections in concurrent lines, use the information from the top of page 296 to come up with a smaller number of reflections in concurrent lines that will accomplish the same thing. Repeat this process. There can be two possible outcomes. Describe them.
- Hint for (b): Given a composition of n reflections in parallel lines, use the result of problem #10 to come up with a smaller number of reflections in parallel lines that will accomplish the same thing. Repeat this process. There can be two possible outcomes. Describe them.
Hint for #18: Consider three non-collinear points whose reflections & rotations are particularly easy to determine. Show that these three points get treated the same way by M_L as they do by (R_(-theta))*(M_k)*(R_theta)

Suggested problems:

6.1 # 1, 2, 4, 7, 8, 10, 11, 13, 17
6.2 # 1, 6, 9, 11, 12, 13, 14, 15

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Last updated March 8, 2009.